Ok, well. I found a solution.
I have only been working with rest for a day and a half now, so I'm not sure it is the BEST solution, but I got it to work.
Basically, I did a POST to backupServers , running a VeeamZip task.
I got a TaskID from the response, which I used to do a GET on /api/tasks
I waited for the show a State of "Finished" and found the UID for the backup session, in an Href.
I pulled out the junk so I was left with a session ID in this format:
Then I was able to do a GET on backupSessions to find the status of the backup task.
As I said, I have only been fiddling with REST for a day and a half, so that may not be the best way to do it.
If there is an easier way to do it, feel free to chime in.